×

On infinite dimensional homogeneous space. (English) Zbl 1474.22009

Summary: In this paper we show that if \(G\) is a locally compact group with \(H\) closed and \(H \geq G\) such that \(\dim G/H < \infty\), then \(G/H\) contains a copy of \(I^{\omega_0} (G/H)\), where \(\omega_0 (G/H) =\) weight of a connected component of \(G/H\), except perhaps when \(\aleph_0 \geq \omega_0(G/H)\geq 2^{\aleph_0}\) [A. A. G. Michael, J. Lie Theory 18, No. 4, 915–917 (2008; Zbl 1160.22004)].

MSC:

22D05 General properties and structure of locally compact groups

Citations:

Zbl 1160.22004
PDFBibTeX XMLCite
Full Text: Link

References:

[1] Bourbaki, N.: Th´eorie des ensembles. Hermann, Paris, 1970. · Zbl 0282.04001
[2] Bourbaki, N.: Topologie G´en´erale, Chapters 1-4. Hermann, Paris, 1971. · Zbl 0249.54001
[3] Bourbaki, N.: Topologie G´en´erale, Chapters 5-10. Hermann, Paris, 1974. · Zbl 0337.54001
[4] Comfort, W. W., L. C. Robertson: Cardinality constraints for pseudocompact and for totally dense subgroups of compact topological groups.Pacific J. Math.,119, no. 2, 1985, 265-286. · Zbl 0592.22005
[5] Comfort, W. W., T. Soundararajan: Pseudocompact group topologies and totally dense subgroups.Pacific J. Math.,100, no. 1, 1982, 61-84. · Zbl 0451.22002
[6] Engelking, R.: General Topology. Berlin, Heldermann, 1989. · Zbl 0684.54001
[7] Gleason, A., R. Palais: On a class of transformation groups.Amer. J. Math.,79, 1957, 631-648. · Zbl 0084.03203
[8] Hofmann, K. H., S. A. Morris: The Structure Of Compact Groups. de Gruyter, Berlin, 2006. · Zbl 1139.22001
[9] Hofmann, K. H., S. A. Morris: Transitive actions of compact groups and topological dimension.J. Algebra,234, 2000, 454-479. · Zbl 0972.22004
[10] Hulanicki, A.: On locally compact topological groups of power of continuum.Fund. Math.,44, 1957, 156-158. · Zbl 0081.26002
[11] Itzkowitz, G., T. S. Wu: The structure of locally compact groups and metrizability. Annals of the New York Academy of Sciences,704, 1993, 164-174. · Zbl 0819.22003
[12] Iwasawa, K.: On some types of topological groups.Ann. Math.,50, 1949, 507-558. · Zbl 0034.01803
[13] Michael, G.: A counter example in the dimension theory of homogeneous spaces of locally compact groups.J. Lie theory,18, no. 4, 2008, 915-917. · Zbl 1160.22004
[14] Montgomery, D., L. Zippin: Topological Transformation Groups. Interscience publishers, New York, 1955. · Zbl 0068.01904
[15] Mostert, P. S.: Sections in principal fiber spaces.Duke Math. J.,23, 1956, 57-72. · Zbl 0072.18102
[16] Skljarenko, E. G.: Homogeneous spaces of an infinite number of dimensions.Soviet Math. Dokl.,2, 1961, 1569-1571. · Zbl 0124.38101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.